EIGEN FUNCTIONS OF DISCRETE-TIME LTI SYSTEMS

What is Lti Systems?

The linearitymeans that the relationship between theinputand theoutputof thesystemis a linear map: If theinput the response and theinput response then n scaled and the summedinput = the response is also scaled and summed. Thelinearityin theinputproduces the correspondingoutputfunction can be represented by theoutputfunctions, scaled and the summed. Thetime invariancemeans that theinputapplied to thesystemat the present orNsamples from the present, theoutputwill be same except for the time delay of theNsamples. This is, if theoutputdue to theinputx[n] isy[n], then the output due to theinputx[n - N] isy[n - N]. So thesystemis thetime invariantbecause the output does not depend on that particular time theinputis applied to thesystem. The fundamental response in theLTI systemtheory is that theLTI systemcan be characterized entirely by the single function called thesystem'simpulse response. Theoutputof thesystemis the convolution of theinputto thesystem. The same response is for thediscrete-time linearshift-invariant systems.

Eigen functions for a discrete time LTI systems

Theeigen functionis the function for which theoutputof the operator of thesystemis the scaled version of the same function. That is,

H f = λ f

Herefis theeigen functionand λ is used to represent theeigen value, which is a constant. The exponential functionsAest. The exponential function is Here Z, N ε C, are theeigen functionsof the lineartime-invariantsystemoperator. T ε R is the sampling interval, and Z, N ε C. To prove it if we take theinputof thesystemisx[n] =. Theoutputof thesystemwith the impulse responseh[n] is then

h[ n] =

This is equivalent to the following by the commutative property of convolution

is dependent only on the parameterz. Soznis the eigen function of an LTIsystembecause thesystemresponse is the same as theinputtimes the constantH[z] applied to the system. In engineering theeigen functionof the linear operator,A, defined on some function space is any non-zero functionfin that space that returns from the operator exactly as is, except for a multiplicative scaling factor.

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EIGEN FUNCTIONS OF DISCRETE-TIME LTI SYSTEMS

What is Lti Systems?

The linearitymeans that the relationship between theinputand theoutputof thesystemis a linear map: If theinput the response and theinput response then n scaled and the summedinput = the response is also scaled and summed. Thelinearityin theinputproduces the correspondingoutputfunction can be represented by theoutputfunctions, scaled and the summed. Thetime invariancemeans that theinputapplied to thesystemat the present orNsamples from the present, theoutputwill be same except for the time delay of theNsamples. This is, if theoutputdue to theinputx[n] isy[n], then the output due to theinputx[n - N] isy[n - N]. So thesystemis thetime invariantbecause the output does not depend on that particular time theinputis applied to thesystem. The fundamental response in theLTI systemtheory is that theLTI systemcan be characterized entirely by the single function called thesystem'simpulse response. Theoutputof thesystemis the convolution of theinputto thesystem. The same response is for thediscrete-time linearshift-invariant systems.

Eigen functions for a discrete time LTI systems

Theeigen functionis the function for which theoutputof the operator of thesystemis the scaled version of the same function. That is,

H f = λ f

Herefis theeigen functionand λ is used to represent theeigen value, which is a constant. The exponential functionsAest. The exponential function is Here Z, N ε C, are theeigen functionsof the lineartime-invariantsystemoperator. T ε R is the sampling interval, and Z, N ε C. To prove it if we take theinputof thesystemisx[n] =. Theoutputof thesystemwith the impulse responseh[n] is then

h[ n] =

This is equivalent to the following by the commutative property of convolution

is dependent only on the parameterz. Soznis the eigen function of an LTIsystembecause thesystemresponse is the same as theinputtimes the constantH[z] applied to the system. In engineering theeigen functionof the linear operator,A, defined on some function space is any non-zero functionfin that space that returns from the operator exactly as is, except for a multiplicative scaling factor.

Ouremail-based homework helpassistance offers brilliant insights and simulations which help make the subject practical and pertinent for anyassignment help.

Transtutors.comprovides timely homework help at reasonable charges with detailed answers to yourEngineeringquestionsso that you get to understand your assignments or homework better apart from having the answers. Our tutors are remarkably qualified and have years of experience providing TheSignals & Systemshomework help or assignment help.